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1. Direct sampling methods for isotropic and anisotropic scatterers with point source measurements

   https://doi.org/10.3934/ipi.2022015  

   Harris, Isaac ; Nguyen, Dinh-Liem ; Nguyen, Thi-Phong ( January 2022 , Inverse Problems and Imaging)

   In this paper, we consider the inverse scattering problem for recovering either an isotropic or anisotropic scatterer from the measured scattered field initiated by a point source. We propose two new imaging functionals for solving the inverse problem. The first one employs a 'far-field' transform to the data which we then use to derive and provide an explicit decay rate for the imaging functional. In order to analyze the behavior of this imaging functional we use the factorization of the near field operator as well as the Funk-Hecke integral identity. For the second imaging functional the Cauchy data is used to define the functional and its behavior is analyzed using the Green's identities. Numerical experiments are given in two dimensions for both isotropic and anisotropic scatterers.

    

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2. A new sampling indicator function for stable imaging of periodic scattering media

   https://doi.org/10.1088/1361-6420/acce5f  

   Nguyen, Dinh-Liem ; Stahl, Kale ; Truong, Trung ( May 2023 , Inverse Problems)

   Abstract This paper is concerned with the inverse problem of determining the shape of penetrable periodic scatterers from scattered field data. We propose a sampling method with a novel indicator function for solving this inverse problem. This indicator function is very simple to implement and robust against noise in the data. The resolution and stability analysis of the indicator function is analyzed. Our numerical study shows that the proposed sampling method is more stable than the factorization method and more efficient than the direct or orthogonality sampling method in reconstructing periodic scatterers. 

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3. Sampling type method combined with deep learning for inverse scattering with one incident wave

   Le, Thu ; Nguyen, Dinh-Liem ; Nguyen, Vu ; Truong, Trung ( January 2023 , AMS Contemporary Mathematics)

   We consider the inverse problem of determining the geometry of penetrable objects from scattering data generated by one incident wave at a fixed frequency. We first study an orthogonality sampling type method which is fast, simple to implement, and robust against noise in the data. This sampling method has a new imaging functional that is applicable to data measured in near field or far field regions. The resolution analysis of the imaging functional is analyzed where the explicit decay rate of the functional is established. A connection with the orthogonality sampling method by Potthast is also studied. The sampling method is then combined with a deep neural network to solve the inverse scattering problem. This combined method can be understood as a network using the image computed by the sampling method for the first layer and followed by the U-net architecture for the rest of the layers. The fast computation and the knowledge from the results of the sampling method help speed up the training of the network. The combination leads to a significant improvement in the reconstruction results initially obtained by the sampling method. The combined method is also able to invert some limited aperture experimental data without any additional transfer training. 

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4. Analysis of the monotonicity method for an anisotropic scatterer with a conductive boundary

   https://doi.org/10.1088/1361-6420/ad7053  

   Harris, Isaac ; Hughes, Victor ; Lee, Heejin ( August 2024 , Inverse Problems)

   In this paper, we consider the inverse scattering problem associated with an anisotropic medium with a conductive boundary. We will assume that the corresponding far–field pattern is known/measured and we consider two inverse problems. First, we show that the far–field data uniquely determines the boundary coefficient. Next, since it is known that anisotropic coefficients are not uniquely determined by this data we will develop a qualitative method to recover the scatterer. To this end, we study the so–called monotonicity method applied to this inverse shape problem. This method has recently been applied to some inverse scattering problems but this is the first time it has been applied to an anisotropic scatterer. This method allows one to recover the scatterer by considering the eigenvalues of an operator associated with the far–field operator. We present some simple numerical reconstructions to illustrate our theory in two dimensions. For our reconstructions, we need to compute the adjoint of the Herglotz wave function as an operator mapping intoH¹of a small ball.

    

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5. Regularized factorization method for a perturbed positive compact operator applied to inverse scattering

   https://doi.org/10.1088/1361-6420/acfd59  

   Harris, Isaac ( October 2023 , Inverse Problems)

   In this paper, we consider a regularization strategy for the factorization method when there is noise added to the data operator. The factorization method is aqualitativemethod used in shape reconstruction problems. These methods are advantageous to use due to the fact that they are computationally simple and require littlea prioriknowledge of the object one wishes to reconstruct. The main focus of this paper is to prove that the regularization strategy presented here produces stable reconstructions. We will show this is the case analytically and numerically for the inverse shape problem of recovering an isotropic scatterer with a conductive boundary condition. We also provide a strategy for picking the regularization parameter with respect to the noise level. Numerical examples are given for a scatterer in two dimensions.

    

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